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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "ground_2/notation/functions/successor_1.ma".
16 include "ground_2/ynat/ynat_pred.ma".
18 (* NATURAL NUMBERS WITH INFINITY ********************************************)
20 (* the successor function *)
21 definition ysucc: ynat → ynat ≝ λm. match m with
26 interpretation "ynat successor" 'Successor m = (ysucc m).
28 lemma ysucc_inj: ∀m:nat. ⫯m = S m.
31 lemma ysucc_Y: ⫯(∞) = ∞.
34 (* Properties ***************************************************************)
36 lemma ypred_succ: ∀m. ⫰⫯m = m.
39 lemma ynat_cases: ∀n:ynat. n = 0 ∨ ∃m. n = ⫯m.
41 [ * /2 width=1 by or_introl/
42 #n @or_intror @(ex_intro … n) // (**) (* explicit constructor *)
43 | @or_intror @(ex_intro … (∞)) // (**) (* explicit constructor *)
47 lemma ysucc_iter_Y: ∀m. ysucc^m (∞) = ∞.
49 #m #IHm whd in ⊢ (??%?); >IHm //
52 (* Inversion lemmas *********************************************************)
54 lemma ysucc_inv_inj: ∀m,n. ⫯m = ⫯n → m = n.
55 #m #n #H <(ypred_succ m) <(ypred_succ n) //
58 lemma ysucc_inv_refl: ∀m. ⫯m = m → m = ∞.
60 #m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
61 #H elim (lt_refl_false m) //
64 lemma ysucc_inv_inj_sn: ∀m2,n1. yinj m2 = ⫯n1 →
65 ∃∃m1. n1 = yinj m1 & m2 = S m1.
67 [ #n1 #H destruct /2 width=3 by ex2_intro/
72 lemma ysucc_inv_inj_dx: ∀m2,n1. ⫯n1 = yinj m2 →
73 ∃∃m1. n1 = yinj m1 & m2 = S m1.
74 /2 width=1 by ysucc_inv_inj_sn/ qed-.
76 lemma ysucc_inv_Y_sn: ∀m. ∞ = ⫯m → m = ∞.
81 lemma ysucc_inv_Y_dx: ∀m. ⫯m = ∞ → m = ∞.
82 /2 width=1 by ysucc_inv_Y_sn/ qed-.
84 lemma ysucc_inv_O_sn: ∀m. yinj 0 = ⫯m → ⊥. (**) (* explicit coercion *)
85 #m #H elim (ysucc_inv_inj_sn … H) -H
89 lemma ysucc_inv_O_dx: ∀m. ⫯m = 0 → ⊥.
90 /2 width=2 by ysucc_inv_O_sn/ qed-.
92 (* Eliminators **************************************************************)
94 lemma ynat_ind: ∀R:predicate ynat.
95 R 0 → (∀n:nat. R n → R (⫯n)) → R (∞) →
97 #R #H1 #H2 #H3 * // #n elim n -n /2 width=1 by/